(α, β)-Derivation on Matrix Ring Mn(R)

Dania Azzahra; Fitriani Fitriani; Ahmad Faisol

doi:10.37905/euler.v14i1.37789

(α, β)-Derivation on Matrix Ring Mn(R)

Dania Azzahra, Fitriani Fitriani, Ahmad Faisol

Abstract

In ring theory, a derivation is an additive mapping d:R→R satisfying Leibniz’s rule. A well-known generalization of this notion is the (α,β)-derivation, defined with respect to two ring endomorphisms α and β. In this paper, we study (α,β)-derivations on the matrix ring Mₙ(R) and several of its subrings, including scalar matrices, diagonal matrices, and upper and lower triangular matrix rings. It is shown that an (α,β)-derivation on the base ring RRR induces an (α′,β′)-derivation on these matrix subrings via entrywise extension, preserving their structural properties. Furthermore, we examine certain properties of (α,β)-derivations on the direct product ring R×R. In particular, we show that the sum of two (α,β)-derivations does not necessarily form an (α,β)-derivation, which is demonstrated through a counterexample.

Keywords

(α, β)–derivation; Matrix ring Mn(R); Triangular matrices; Direct product ring

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References

I. Ernanto, “Sifat-sifat ring faktor yang dilengkapi derivasi,” Journal of Fundamental Mathematics and Applications, vol. 1, no. 1, pp. 12–21, 2018, doi: 10.14710/jfma.v1i1.3.

A. B. Thomas, N. P. Puspita, and Fitriani., “Derivation on several rings.” Barekeng: Jurnal Ilmu Matematika dan Terapan, vol. 18, no. 3, pp. 1729–1738, 2024, doi: 10.30598/barekengvol18iss3pp1729-1738.

M. Bresar and J. Vukman, “Jordan derivations on prime rings,” Bulletin of the Australian Mathematical Society, vol. 37, no. 3, pp. 321–322, 1988, doi: 10.1017/S0004972700026927.

A. Ber, K. Kudaybergenov, and F. Sukochev, “Notes on derivations of murray–von neumann algebras,” Journal of Functional Analysis, vol. 279, no. 1, pp. 1–26, 2020, doi: 10.1016/j.jfa.2020.108589.

K. Hattori and H. Kojima, “Rings of nilpotent elements of monomial derivations on polynomial rings,” Communications in Algebra, vol. 52, no. 7, pp. 2998–3009, 2024, doi: 10.1080/00927872.2024.2312459.

D. Liu and X. Sun, “Retracts that are kernels of locally nilpotent derivations,” Czechoslovak Mathematical Journal, vol. 72, no. 1, pp. 191–199, 2022, doi: 10.21136/CMJ.2021.0388-20.

X. Sun and B. Wang, “Images of locally nilpotent derivations of bivariate polynomial algebras over a domain,” Czechoslovak Mathematical Journal, vol. 74, no. 2, pp. 599–610, 2024, doi: 10.21136/CMJ.2024.0008-24.

D. L. Mursyidah, B. H. S. Utami, Fitriani, and A. Faisol, “Nil derivation and δ-ideal on polynomial ring,” Barekeng: Jurnal Ilmu Matematika dan Terapan, vol. 19, no. 3, pp. 1069–1078, 2025, doi: 10.30598/barekengvol20iss1pp0325-0334.

R. Waluyo, A. Faisol, and Fitriani, “(σ, τ)-derivasi pada ring grup,” Euler: Jurnal Ilmiah Matematika, Sains dan Teknologi, vol. 13, no. 2, pp. 142–146, 2025, doi: 10.37905/euler.v13i2.31564.

S. K. Ekrami, “Jordan higher derivations: A new approach,” Journal of Algebraic Systems, vol. 10, no. 1, pp. 167–177, 2022.

B. Bhushan, G. S. Sandhu, S. Ali, and D. Kumar, “On centrally extended jordan derivations and related maps in rings,” Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 1, pp. 23–35, 2023, doi: 10.15672/hujms.1008922.

N. Bera and B. Dhara, “Jordan homoderivation behavior of generalized derivations in prime rings,” Ukrains’kyi Matematychnyi Zhurnal, vol. 75, no. 9, pp. 1178–1194, 2023, doi: 10.1007/s11253-024-02265-3.

A. Ma, L. Chen, and Z. Qin, “Jordan semi-triple derivations and jordan centralizers on generalized quaternion algebras,” AIMS Mathematics, vol. 8, no. 3, pp. 6026–6035, 2023, doi: 10.3934/math.2023304.

D. Ren and J. Zhang, “Nonlinear mixed ∗-jordan type higher derivations on ∗-algebras,” Filomat, vol. 39, no. 15, pp. 5203–5216, 2025, doi: 10.2298/FIL2515203R.

A. S. Alali, H. M. Alnoghashi, and N. u. Rehman, “Centrally-extended jordan ∗-derivations centralizing symmetric or skew elements,” Axioms, vol. 12, no. 1, p. 86, 2023, doi: 10.3390/axioms12010086.

G. N. Malleswari and S. Sreenivasulu, “On skew jordan product and generalized derivations in prime rings with involution,” JP Journal of Algebra, Number Theory and Applications, vol. 63, no. 4, pp. 329–334, 2024, doi: 10.17654/0972555524020.

M. A. Madni, M. R. Mozumder, A. Z. Ansari, and F. Shujat, “Characterization of multiplicative mixed jordan-type derivations on rings with involution,” European Journal of Pure and Applied Mathematics, vol. 18, no. 2, 2025, doi: 10.29020/nybg.ejpam.v18i2.6084.

F. Shujat, F. Alharbi, and A. Z. Ansari, “Weak (p, q)-jordan centralizer and derivation on rings and algebras,” AIMS Mathematics, vol. 10, no. 4, pp. 8322–8330, 2025, doi: 10.3934/math.2025383.

S. Ali, N. N. Rafiquee, and V. Varshney, “Certain types of derivations in rings: A survey,” Journal of the Indonesian Mathematical Society, vol. 30, no. 2, pp. 256–306, 2024, doi: 10.22342/jims.30.2.1623.256-306.

W. Ahmed, A. S. Alali, and M. R. Mozumder, “Characterization of (α, β)-jordan bi-derivations in prime rings,” AIMS Mathematics, vol. 9, no. 6, pp. 14549–14557, 2024, doi: 10.3934/math.2024707.

N. A. Syaharani, Fitriani, S. L. Chasanah, and A. Faisol, “(α′, β′)-derivation on the polynomial ring K[x],” Journal of the Indonesian Algebra Society, vol. 1, no. 1, pp. 17–29, 2026, [Online]. Available: https://indoas.id/journal/index.php/JIAS/article/view/jias.17.29.2026.

Fitriani, I. E. Wijayanti, A. Faisol, and S. Ali, “On f-derivations on polynomial modules,” Journal of Algebra and Its Applications, vol. 21, no. 12, p. 2250235, 2022, doi: 10.1142/S0219498825501555.

A. Faisol and Fitriani, “A study of derivations and linear mappings on skew generalized power series modules,” Barekeng: Jurnal Ilmu Matematika dan Terapan, vol. 19, no. 4, pp. 3047–3058, 2025, doi: 10.30598/barekengvol19iss4pp3047-3058.

S. Wahyuni, I. E. Wijayanti, D. A. Yuwaningsih, and A. D. Hartanto, Teori Ring dan Modul. Gadjah Mada University Press, 2017, [Online]. Available: https://digitalpress.ugm.ac.id/book/346.

DOI: https://doi.org/10.37905/euler.v14i1.37789